Best Known (43−11, 43, s)-Nets in Base 32
(43−11, 43, 209717)-Net over F32 — Constructive and digital
Digital (32, 43, 209717)-net over F32, using
- 321 times duplication [i] based on digital (31, 42, 209717)-net over F32, using
- net defined by OOA [i] based on linear OOA(3242, 209717, F32, 11, 11) (dual of [(209717, 11), 2306845, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3242, 1048586, F32, 11) (dual of [1048586, 1048544, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(3241, 1048577, F32, 11) (dual of [1048577, 1048536, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(3233, 1048577, F32, 9) (dual of [1048577, 1048544, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(3242, 1048586, F32, 11) (dual of [1048586, 1048544, 12]-code), using
- net defined by OOA [i] based on linear OOA(3242, 209717, F32, 11, 11) (dual of [(209717, 11), 2306845, 12]-NRT-code), using
(43−11, 43, 1048590)-Net over F32 — Digital
Digital (32, 43, 1048590)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3243, 1048590, F32, 11) (dual of [1048590, 1048547, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(3241, 1048576, F32, 11) (dual of [1048576, 1048535, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(3229, 1048576, F32, 8) (dual of [1048576, 1048547, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(322, 14, F32, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
(43−11, 43, large)-Net in Base 32 — Upper bound on s
There is no (32, 43, large)-net in base 32, because
- 9 times m-reduction [i] would yield (32, 34, large)-net in base 32, but